Rough Wetting

EUROPHYSICS LETTERS 15 July 2001 Europhys. Lett., 55 (2), pp. 214–220 (2001)

J. Bico, C. Tordeux and D. Quer´ e´ Laboratoire de Physique de la Matière Condensée, URA 792 du CNRS Collège de France - 75231 Paris Cedex 05, France

(received 25 January 2001; accepted 9 May 2001)

PACS. 68.03.-g – Gas-liquid and vacuum-liquid interfaces. PACS. 68.35.Ct – Interface structure and roughness. PACS. 68.08.Bc – Wetting.

Abstract. – If a rough surface is put in contact with a wetting liquid, the roughness may be spontaneouslyinvaded depending on the surface pattern and the wetting properties of the liquid. Here, we studythe conditions for observing such an imbibition and present practical achievements where the wetting properties of the surface can be predicted and tuned bythe design of a solid texture. The contact angle of a drop on such a surface (where solid and liquid coexist) is discussed. Finally, the dynamics of the liquid ﬁlm is found to obey a diﬀusive-type law, as in the case of porous wicking.

Introduction. – The wetting behavior of an ideal flat solid is fixed by its chemical composition (Young’s relation). But real solids are rough, which affects their wettability (or non-wettability) [1–3]. For example, if hydrophobicity and roughness are cleverly combined, a water drop deposited on such a surface can remain nearly spherical [4–7]. This can be understood by the fact that the liquid is only in contact with the upper part of the relief: the roughness is mainly filled with air, which leads to super-hydrophobicity [6,7]. This letter deals with the opposite situation where the surface is rather hydrophilic. We first point out a possible invasion of the roughness, when a textured surface is put in contact with a liquid reservoir. The conditions of this imbibition are described as a function of the geometric parameters of the surface design. Then, we study the apparent contact angle of a drop deposited on a textured surface, and finally characterize the dynamics of invasion. Critical contact angle of imbibition. – Figure 1 shows a rough surface put in contact with a liquid reservoir, with a film propagating inside the texture. The Young relation is locally valid and for partial wetting the top of the roughness remains dry. We call φS the solid fraction remaining dry, and r the solid roughness (ratio of the actual solid area over its projected one). If the imbibition front rises by a small quantity dz, the interfacial energies change with a quantity dE (written per unit width of the sample) given by

dE =(γSL − γSV)(r − φS)dz + γ(1 − φS)dz, (1) c EDP Sciences J. Bico et al.: Rough wetting 215

Fig. 1 Fig. 2

Fig. 1 – Rough surface in contact with a reservoir. A small displacement dz of the imbibition front of height z is considered.

Fig. 2 – Microstructured surface with regular micronic spikes used for the experiment. Surface parameters can accuratelybe estimated from the picture. We get r =1.3andφS =5%.

where γSL, γSV and γ are, respectively, the solid/liquid, solid/vapor and liquid/vapor interfacial tensions. The ﬁrst term in eq. (1) is related to the replacement of a dry solid by a wet one. It is proportional to the wetted area, i.e. to the factor (r − φS). The second term is less usual in impregnation processes: it is related to the creation of a liquid/vapor interface associated with the ﬁlm propagation. Note that gravity was ignored in (1), which corresponds to textures of small size (much smaller than the capillary length) and small heights. The gravity-limited case is treated in the appendix. The liquid should rise if dE is negative. Introducing Young’s law (γ cos θ = γSV − γSL) gives, as a condition for imbibition,

1 − φS θ<θc with cos θc = , (2) r − φS where θ is the equilibrium contact angle of the liquid on an ideal flat surface of the same chemical composition. Criterion (2) (logically) appears as intermediate between wetting and wicking criteria. For a flat surface (r → 1), the surface is wetted if the contact angle reaches 0 (θc = 0), while a porous medium (r →∞) is invaded for liquids having a contact angle smaller than π/2 (θc = π/2). More generally, since we have r>1andφS < 1, eq. (2) always defines a critical contact angle intermediate between 0 and π/2. Thus, the ability of a textured surface to drive a liquid can be tuned by its surface design. For a given surface composition and liquid (i.e., fixing θ), the nature of the texture (which determines r and φS) decides if condition (2) is satisfied, or not. In a general case (disordered surfaces), the parameters r and φS are deeply intricate (and φS may depend on θ). However, the use of micropatterned surfaces allows to decouple these two parameters, and even to treat them as independent, which is now discussed using such a model surface. 216 EUROPHYSICS LETTERS

Fig. 3 Fig. 4

Fig. 3 – Pinning of the contact lines on the corners of the crenellations which ensures a constant value for the surface fraction φS.

Fig. 4 – Metastable square drop of ethyl malonate (θ =32◦) on the structured surface. The contact line follows the directions of the rows.

Surfaces with a controlled design. – Figure 2 shows a surface achieved by Marzolin by casting a sol-gel silicate on a silicon wafer [8]. It consists in spikes (typical size one micron) which are regularly spaced. We used this model surface to test a possible invasion of the texture by various liquids. For having a partial wetting with oils, octadecyl trichlorosilanes were grafted on the sample. A planar silicon wafer where the same molecules were grafted was used as a reference surface, allowing us to measure the advancing contact angle θ. The measurements were done optically with an accuracy of 5◦ and the contact angle hysteresis on these surfaces was found to be small (< 10◦). Then, experiments similar to the one sketched in fig. 1 were realized with different liquids (i.e. various contact angles). The sample was partially immersed in a reservoir of liquid, and the existence of an invasion (or not) was monitored, together with the dynamics of the rise. The film was easily detected thanks to the darkening it induced on the sample. Our major result is the observation of an invading film climbing up to the top of the sample (typically one centimeter) for liquids having a small contact angle. A threshold in contact angle was indeed observed, above which the invasion was not observed. The value of this threshold was found to be between 30◦ and 35◦. The surface roughness r is easy to estimate from SEM (scanning electronic microscopy) images and is r =1.30 (±0.05). The solid fraction φS on such a surface should be independent of the contact angle θ, because of the possibility for the contact lines to pin on the corners of the crenellations (fig. 3) [9]. If the angle is larger than 0, the top of the spikes remains dry; in the same time, an angle smaller than π/2 makes favorable the filling of the texture by the liquid. In such conditions, the parameter φS is just the ratio of the area of the top of the spikes over the total area of the sample, which is 5% here [10]. Then, relation (2) predicts a ◦ value of θc equal to 40 . This value is close to the observed one—but we consider as significant the small difference between both. Criterion (2) is thermodynamic, and metastable states can also be achieved because there are activation barriers to jump for the contact line. This is related to the disconnected nature of the texture: if we easily understand how a wetting liquid can invade a continuous groove in a solid [11], it is less obvious to figure out how the contact line pinned on a row of spikes can “know” that there is another row ahead. (Thus, hysteresis can occur: J. Bico et al.: Rough wetting 217

Fig. 5 – a) Wenzel regime: the surface is dryahead the contact line. b) Film regime: the roughness is filled with a liquid film. The drop lies on a solid/liquid composite surface. it is not the same thing to force a film inside a textured solid and leave it evolve, and to let it spontaneously invade the textures.) In our case, the progression of a film could be ensured by the presence of menisci around each spike and conditioned by the fact that each meniscus can reach the spike ahead from it. If we simplify the meniscus in a one-dimensional prism, the minimum contact angle θmin ensuring a sufficient extension is simply given by tan θmin = δ/d, with d the distance between two spikes and δ their height. For our sample (δ =1.2 µmand ◦ d =2.5 µm), θmin is equal to 27 , slightly lower than θc, as observed experimentally. If a drop is deposited in a metastable situation (θmin <θ<θc), the liquid contact line follows the spikes rows and adopts a square shape as illustrated in fig. 4. This result reminds beautiful experiments from Lenormand on capillary imbibition of two-dimensional porous networks [12]. A symmetric situation occurs when one tries to mop up a liquid drop deposited on the surface: dewetting is expected if the Young contact angle is higher than the critical value ◦ (θc =40 in case). A much higher value was experimentally observed: after the drop has been sucked up, a film remains trapped if θ is lower than 60◦. As it recedes, the imbibition front has to dewet spike rows. On such rows, both the solid roughness and the surface fraction are locally higher than their mean value: we measure for a row r =2.0andφS = 20%, which ◦ gives from eq. (2) θc =63 , in much better agreement with our measurements. Effective contact angle. – A related question is the contact angle adopted by a small drop deposited on a rough surface. Two regimes are expected depending on whether the roughness is flooded or not (figs. 5a) and b)). 1) If the Young contact angle θ exceeds the critical angle θc, the rough surface is dry ahead the contact line (fig. 5a)). This situation has been described by Wenzel [1] and leads to a magnification of the cosine of the contact angle:

cos θ∗ = r cos θ, (3) where θ∗ is the eﬀective contact angle on the rough surface. 2) When the Young angle is smaller than θc, the roughness is impregnated (then, part of the liquid is sucked, which can generally be neglected because of the small size of the texture) (ﬁg. 5b)). Thus, the drop stands on a surface composed with solid and liquid. Considering a small variation dx of the position of the contact line leads to a variation of surface energy ∗ dE =(γSL − γSV)φS dx − γ(1 − φS)dx + γ cos θ dx. The latter quantity is zero at equilibrium, which leads to (using Young’s relation)

∗ cos θ =1− φS(1 − cos θ). (4)

This expression shows that the effect of a film is indeed to improve the wetting (θ∗ <θ), but that it is not possible to induce a wetting transition by texturing a solid: complete wetting of the rough surface (θ∗ = 0) is only achieved when θ = 0. The crossover between the two regimes is continuous (eqs. (3) and (4) are equivalent for θ = θc), which is a difference with 218 EUROPHYSICS LETTERS

Fig. 6 Fig. 7

Fig. 6 – Cosine of the contact angle θ∗ measured on the textured surface as a function of the angle θ on the reference surface. The emptyand full symbols correspond to static receding and advancing angles, respectively, and the full lines to eqs. (3) and (4).

Fig. 7 – Rise experiment realized with a silicon oil (γ =20.6mN/m, θ =0,η = 16 mPa s). The front height is plotted vs. the square root of the time. The dotted line is eq. (5) with β =4.1. the hydrophobic case, where the contact angle is observed to jump as soon as air is trapped below the drop [2,6,13]. Drops of various liquids were deposited on both the reference and the textured surfaces. The cosine of the observed contact angle θ∗ on the rough surface is displayed in fig. 6 as a function of the cosine of the angle θ on the reference surface. The lines corresponding to eqs. (3) and (4) have been drawn without any adjustable parameter and show a fair agreement with experimental data. The range of the imbibition regime in fig. 6 is rather small because of the sample mi- crostructure and it would be worth realizing other designs with various values of r and φS. But it is indeed observed in this regime that the contact angle remains non-zero, in a domain of angle θ where Wenzel’s law would have predicted a wetting transition (θ∗ = 0). Note finally that the Wenzel law seems to remain valid when θ is slightly higher than 90◦ (the drop weight is probably sufficient to overcome a weak pinning of the contact line on the corner of crenellations). Dynamics of imbibition. – The time evolution of the imbibition front is a major issue in practical applications. It reminds capillary rise in porous materials, which is generally described by the classical Washburn law [14]: a constant capillary pressure balances a Darcy viscous dissipation, which leads to a diffusion law for the imbibition. Here, the driving capillary pressure ∆PL derives from the relation dE = −∆PL dV , where dE is the energy variation induced by a small rise dz of the front line (eq. (1)) and dV the associated variation of liquid volume. For crenellated surfaces, dV (written per unit length) is a simple function of φS and δ, the depth of the roughness: dV =(1− φS)δ dz.Thus,we find: ∆PL = γ/δ(cos θ − cos θc)/ cos θc. The viscous force per unit area in the liquid film can be estimated from a classical Poiseuille law for a liquid flowing on a plane. It writes: 2 ∆PV =3ηV z/δ , noting η the liquid viscosity and V its mean velocity (V =dz/dt). The J. Bico et al.: Rough wetting 219 presence of the spikes should not alter the scaling laws in this force, but increase it by a numerical factor that we call β. If hydrostatic pressure is negligible (which is the case at the beginning of the imbibition), balancing the capillary pressure with the viscous force leads to a diffusion law

1/2 2 cos θ − cos θc γδ z = t . (5) 3β cos θc η

Gravity would soften the flow rate and finally provide an equilibrium height (of metric scale) described in appendix. Rise experiments were recorded with a video camera. The front line is slightly distorted by defects on the sample but the data are reproducible and the rise rate does not depend on the surface orientation. An experiment made with a silicone oil (γ =20.6mN/m, θ =0◦, η = 16 cP) is diplayed in fig. 7. The rise height z is plotted vs. the square root of time. A diffusive-type law is indeed observed, as expected. The zero ordinate corresponds to the millimetric extension of the macroscopic meniscus. The slope of the straight line allows us to estimate the coefficient β for our surface: β =4.1, which was confirmed with other wetting liquids. The validity of eq. (5) was finally checked in a partial wetting situation (for θ<θc). The imbibition was still found to be diffusive but the rise slown down because of the reduction in the driving force due to the non-zero value of θ. The contact angles deduced from a fit with eq. (5) were found to be close (less than 5◦) to the ones measured optically on the reference surface. Conclusion. – We have described and illustrated with a model surface the different regimes of wetting of a rough surface exposed to a liquid. A two-dimensional imbibition of the roughness is observed if the Young contact angle of the liquid (obtained on an ideal smooth surface of same composition) is below a critical value θc, which depends on two independent geometrical parameters: the surface roughness r (ratio of the actual surface over the projected one) and the solid fraction φS (surface fraction corresponding to the top of the relief). The dynamics of imbibition film follows a diffusive-type law, as classically observed in wicking of a three-dimensional porous medium. Thus, surface roughness modifies the contact angle when a drop is deposited on the surface. Beyond θc, the liquid explores the whole relief and the cosine of the liquid contact angle is magnified by the surface roughness (Wenzel regime). Below θc (imbibition regime), the drop lays on a composite solid/liquid surface which leads to a large reduction of the contact angle. Building other micropatterned surfaces should be helpful for generalization of this approach to more complex structures. It should help to control the wetting and imbibition properties of a surface by tuning its design. ∗∗∗ We thank S. Herminghaus for sending to us the very similar results he obtained for the criterion of impregnation in a texture (eq. (2)). We also thank C. Marzolin for putting at our disposal his model surfaces.

Appendix

The law of capillary rise for a textured surface. – Hydrostatic pressure limits the rise in ﬁg. 1. Balancing gravity with the capillary pressure ∆PL leads to the equilibrium height h = γ/ρgδ(cos θ − cos θc)/ cos θc, with ρ the liquid density. Thus, heights of some meters are 220 EUROPHYSICS LETTERS expected for micrometric patterns, which justify that we neglect gravity for centimetric rises as presented in the section Dynamics of imbibition. Imbibition might be followed by weighing the sample while the liquid is rising. Two forces are then detected: a capillary suction fc by the macroscopic meniscus and the weight fw of the liquid exploring the rough structure. The capillary force (per unit length of the contact line) ∗ writes fc = γ cos θ . The weight of liquid at the height z is easily evaluated for crenellated structures (also per unit length): fw =(1− φS)zδρg. Once the equilibrium height is reached (z = h), the global force f measured is given by f = rγ cos θ, which surprisingly provides a simple measurement of the surface roughness r. Practically, it would require long samples (or larger spikes), which are not easy to build.

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